Optimal dimensioning for the corner combined footings

Advances in Computational Design, Vol. 2, No. 2 (2017) 169-183

DOI: https://doi.org/10.12989/acd.2017.2.2.169 169

Copyright © 2017 Techno-Press, Ltd.

http://www.techno-press.org/?journal=acd&subpage=7 ISSN: 2283-8477 (Print), 2466-0523 (Online)

Optimal dimensioning for the corner combined footings

Sandra López-Chavarríaa, Arnulfo Luévanos-Rojas* and Manuel Medina-Elizondob

Institute of Multidisciplinary Researches, Autonomous University of Coahuila, Blvd. Revolución No. 151 Ote, CP

27000, Torreón, Coahuila, México

(Received December 5, 2016, Revised March 10, 2017, Accepted March 15, 2017)

Abstract. This paper shows optimal dimensioning for the corner combined footings to obtain the most

economical contact surface on the soil (optimal area), due to an axial load, moment around of the axis “X”

and moment around of the axis “Y” applied to each column. The proposed model considers soil real

pressure, i.e., the pressure varies linearly. The classical model is developed by trial and error, i.e., a

dimension is proposed, and after, using the equation of the biaxial bending is obtained the stress acting on

each vertex of the corner combined footing, which must meet the conditions following: 1) Minimum stress

should be equal or greater than zero, because the soil is not withstand tensile. 2) Maximum stress must be

equal or less than the allowable capacity that can be capable of withstand the soil. Numerical examples are

presented to illustrate the validity of the optimization techniques to obtain the minimum area of corner

combined footings under an axial load and moments in two directions applied to each column.

Keywords: corners combined footings; optimization techniques; contact surface; more economical

dimension; optimal area

1. Introduction

Footings are structural elements that transmit column or wall loads to the underlying soil below

the structure. Footings are designed to transmit these loads to the soil without exceeding its safe

bearing capacity, to prevent excessive settlement of the structure to a tolerable limit, to minimize

differential settlement, and to prevent sliding and overturning. The choice of suitable type of

footing depends on the depth at which the bearing stratum is localized, the soil condition and the

type of superstructure. The foundations are classified into superficial and deep, which have

important differences: in terms of geometry, the behavior of the soil, its structural functionality

and its constructive systems (Bowles 2001, Das et al. 2006).

The design of superficial solution is done for the following load cases: 1) the footings subjected

to concentric axial load, 2) the footings subjected to axial load and moment in one direction

(uniaxial bending), 3) the footings subjected to axial load and moment in two directions (biaxial

bending) (Bowles 2001, Das et al. 2006, Calabera 2000, Tomlinson 2008, McCormac and Brown

Corresponding author, Ph.D., E-mail: [email protected] aPh.D., E-mail: [email protected]

bPh.D., E-mail: [email protected]

mailto:[email protected]



Sandra López-Chavarría, Arnulfo Luévanos-Rojas and Manuel Medina-Elizondo

2013, González-Cuevas and Robles-Fernandez-Villegas 2005).

A combined footing is a long footing supporting two or more columns in (typically two) one

row. The combined footing may be rectangular, trapezoidal or T-shaped in plan. Rectangular

footing is provided when one of the projections of the footing is restricted or the width of the

footing is restricted. Trapezoidal footing or T-shaped is provided when one column load is much

more than the other. As a result, both projections of the footing beyond the faces of the columns

will be restricted (Kurian 2005, Punmia et al. 2007, Varghese 2009).

Construction practice may dictate using only one footing for two or more columns due to:

a) Closeness of column (for example around elevator shafts and escalators).

b) To property line constraint, this may limit the size of footings at boundary. The eccentricity

of a column placed on an edge of a footing may be compensated by tying the footing to the interior

column.

Conventional method for design of combined footings by rigid method assumes that (Bowles

2001, Das et al. 2006, McCormac and Brown 2013, González-Cuevas and Robles-Fernandez-

Villegas 2005):

1. The footing or mat is infinitely rigid, and therefore, the deflection of the footing or mat does

not influence the pressure distribution.

2. The soil pressure is linearly distributed or the pressure distribution will be uniform, if the

centroid of the footing coincides with the resultant of the applied loads acting on foundations.

3. The minimum stress should be equal to or greater than zero, because the soil is not capable

of withstand tensile stresses.

4. The maximum stress must be equal or less than the allowable capacity that can withstand the

soil.

Optimization of building structures is a prime target for designers and has been investigated by

many researchers in the past and its papers are: Optimum design of unstiffened built-up girders

(Ha 1993); Shape optimization of RC flexural members (Rath et al. 1999); Sensitivity analysis and

optimum design curves for the minimum cost design of singly and doubly reinforced concrete

beams (Ceranic and Fryer 2000); Optimal design of a welded I-section frame using four

conceptually different optimization algorithms (Jarmai et al. 2003); New approach to optimization

of reinforced concrete beams (Leps and Sejnoha 2003); Cost optimization of singly and doubly

reinforced concrete beams with EC2-2001 (Barros et al. 2005); Cost optimization of reinforced

concrete flat slab buildings (Sahab et al. 2005); Multi objective optimization for performance-

based design of reinforced concrete frames (Zou et al. 2007); Design of optimally reinforced RC

beam, column, and wall sections (Aschheim et al. 2008); Optimum design of reinforced concrete

columns subjected to uniaxial flexural compression (Bordignon and Kripka 2012); A hybrid CSS

and PSO algorithm for optimal design of structures (Kaveh and Talatahari 2012); Structural

optimization and proposition of pre-sizing parameters for beams in reinforced concrete buildings

(Fleith de Medeiros and Kripka 2013); Optimum cost design of RC columns using artificial bee

colony algorithm (Ozturk and Durmus 2013); Optimization of a sandwich beam design: analytical

and numerical solutions (Awad 2013); Cold-formed steel channel columns optimization with

simulated annealing method (Kripka and Chamberlain Pravia 2013); Cost optimization of

reinforced high strength concrete T-sections in flexure (Tiliouine and Fedghouche 2014); Optimal

design of reinforced concrete plane frames using artificial neural networks (Kao and Yeh 2014);

Reliability-based design optimization of structural systems using a hybrid genetic algorithm

(Abbasnia et al. 2014); Numerical experimentation for the optimal design of reinforced rectangular

concrete beams for singly reinforced sections (Luévanos-Rojas 2016a).

170


The papers for optimal design of reinforced concrete foundations are: flexural strength of

square spread footing (Jiang 1983); Closure to “Flexural strength of square spread footing” by Da

Hua Jiang (Jiang 1984); Flexural limit design of column footing (Hans 1985); Economic design

optimization of foundation (Wang and Kulhawy 2008); Reliability-Based Economic design

optimization of spread foundation (Wang 2009); Structural cost of optimized reinforced concrete

isolated footing (Al-Ansari 2013); Multi-objective optimization of foundation using global-local

gravitational search algorithm (Khajehzadeh et al. 2014).

Some papers presenting the equations to obtain the dimension of footings are: A mathematical

model for dimensioning of footings rectangular (Luévanos-Rojas 2013); A mathematical model for

dimensioning of footings square (Luévanos-Rojas 2012a); A mathematical model for the

dimensioning of circular footings (Luévanos-Rojas 2012b); A new mathematical model for

dimensioning of the boundary trapezoidal combined footings (Luévanos-Rojas 2015); A

mathematical model for the dimensioning of combined footings of rectangular shape (Luévanos-

Rojas 2016b).

This paper shows optimal dimensioning for the corner combined footings to obtain the most

economical contact surface on the soil (optimal area), due to an axial load, moment around of the

axis “X” and moment around of the axis “Y” applied to each column. The proposed model

considers soil real pressure, i.e., the pressure varies linearly. The classical model is developed by

trial and error, i.e., a dimension is proposed, and after, using the equation of the biaxial bending is

obtained the stress acting on each vertex of the corner combined footing, which must meet the

conditions following: 1) Minimum stress should be equal or greater than zero, because the soil is

not withstand tensile. 2) Maximum stress must be equal or less than the allowable capacity that can

be capable of withstand the soil. The paper presents numerical examples for two property lines

adjacent to illustrate the validity of the optimization techniques to obtain the minimum area of the

corner combined footings under an axial load and moments in two directions applied to each

column.

2. Formulation of the proposed model

The general equation for any type of footings subjected to bidirectional bending (Luévanos-

Rojas 2012a, b, 2013, 2015, 2016b, Gere and Goodno 2009)

(1)

Where: σ is the stress exerted by the soil on the footing (soil pressure), A is the contact area of

the footing, P is the axial load applied at the center of gravity of the footing, Mx is the moment

around the axis “X”, My is the moment around the axis “Y”, x is the distance in the direction “X”

measured from the axis “Y” to the fiber under study, y is the distance in direction “Y” measured

from the axis “X” to the farthest under study, Iy is the moment of inertia around the axis “Y” and Ix

is the moment of inertia around the axis “X”.

Fig. 1 shows a corner combined footing under axial load and moment in two directions (biaxial

bending) in each column, the pressure below the footing vary linearly (Luévanos-Rojas 2012a, b,

2013, 2015, 2016b).

Fig. 2 presents the pressure diagram below the corner combined footing, and also the stresses in

each vertex are shown.

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Fig. 1 Corner combined footing

Fig. 2 Diagram of pressure below the footing

From the Eq. (1) the stresses in each vertex of footing are obtained

(2)

(3)

(4)

(5)

(6)

172


(7)

Where R is resultant force, MxT is resultant moment around the axis “X ” and MyT is resultant moment

around the axis “Y ” are obtained

(8)

(9)

(10)

The geometric properties of section are

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Geometry conditions are

(18)

(19)

173


(20)

(21)

Substituting the Eqs. (12)-(15) into Eqs. (9)-(10) to obtain the moments in function of “a”, “b”,

“b1” and “b2”, these are

(22)

(23)

Substituting the Eqs. (11) to (17) into Eqs. (2) to (7) to find the stresses in function of “a”, “b”,

“b1” and “b2”, these are

(24)

(25)

(26)

(27)

(28)

(29)

174


The stresses generated by soil on contact surface of the combined footing must meet the

following conditions: 1) The minimum stress should be equal or greater than zero; 2) The

maximum stress must be equal or less than the soil allowable load capacity “σadm” (Bowles 2001,

Das et al. 2006, McCormac and Brown 2013, González-Cuevas and Robles-Fernandez-Villegas

2005).

Now the objective function to minimize the total area of the contact surface “At” is

(30)

Constraint functions for dimensioning of the corner combined footings are

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

175


(40)

(41)

(42)

3. Numerical problems

Tables present five cases for dimensioning of corner combined footings with two boundaries

adjacent; each case varies the soil allowable load capacity of “σadm=250, 225, 200, 175, 150

kN/m2”, and each case presented four types that are same in all cases. Table 1 presented the four

types of corner combined footings; the types vary in the mechanical elements acting on the

footing.

Table 1 Mechanical elements acting on the footing

Type Loads of the column 1 Loads of the column 2 Loads of the column 3

P1 kN Mx1 kN-m My1 kN-m P2 kN Mx2 kN-m My2 kN-m P3 kN Mx3 kN-m My3 kN-m

1 500 150 200 1000 300 200 900 200 250

2 500 -150 200 1000 -300 200 900 -200 250

3 500 150 -200 1000 300 -200 900 200 -250

4 500 -150 -200 1000 -300 -200 900 -200 -250

Tables 2, 3 and 4 presented the results and it make the following considerations: 1) Dimensions

of the three columns are of 40×40 cm in all cases; 2) Soil allowable load capacity varies for each

case; 3) Distance between columns is L1=5.00 m, L2=6.00 m.

Tables 2, 3 and 4 show the results using the optimization techniques; the objective function

(minimum contact surface) by Eq. (30) is obtained, and constraint functions by Eqs. (31) to (42)

are found, and the minimum areas and dimensions for corner combined footings are obtained using

the MAPLE-15 software, and it is assumed that dimensions are nonnegative.

This problem assumes that the constant parameters are: P1, Mx1, My1, P2, Mx2, My2, P3, Mx3, My3,

c1, c2, c3, c4, L1, L2, σadm, R, and the decision variables are: MxT, MyT, a, b, b1, b2, σ1, σ2, σ3, σ4, σ5, σ6.

Table 2 makes the following considerations: R=2400 kN, MxT and MyT are not constrained, 5.40

m≤a, 6.40 m≤b, 0≤b1, 0≤b2, At is objective function, 0≤σ1≤σadm, 0≤σ2≤σadm, 0≤σ3≤σadm, 0≤σ4≤σadm,

0≤σ5≤σadm, 0≤σ6≤σadm. Table 3 makes the following considerations: R=2400 kN, MxT and MyT are

not constrained, 5.40 m≤a, 6.40 m≤b, 1.00 m≤b1, 1.00 m≤b2, At is objective function, 0≤σ1≤σadm,

0≤σ2≤σadm, 0≤σ3≤σadm, 0≤σ4≤σadm, 0≤σ5≤σadm, 0≤σ6≤σadm. Table 4 makes the following

176


considerations: R=2400 kN, MxT and MyT are not constrained, 5.40 m=a, 6.40 m=b, 0≤b1, 0≤b2, At

is objective function, 0≤σ1≤σadm, 0≤σ2≤σadm, 0≤σ3≤σadm, 0≤σ4≤σadm, 0≤σ5≤σadm, 0≤σ6≤σadm.

Tables 2, 3 and 4 are shown in the appendix.

4. Results

Table 2 shows the following results (dimensions a, b, b1 and b2 are assumed nonnegative): 1)

The minimum area is the same for the four types in each case; 2) If the soil allowable load capacity

decreases, the minimum area increases; 3) The resultant mechanical elements are R=2400 kN,

MxT=0, MyT=0 for all cases; 4) The stresses generated by loads in each vertex are the same to σadm

in each case. It means that the resultant force is located in the center of gravity of the contact area

of the footing with soil, i.e., the total eccentricity of the resultant force “R” in the two directions is

zero.

Table 3 presents the following results (dimensions a and b are assumed nonnegative, and b1 and

b2 are greater than or equal to one): 1) The minimum area is deferent for the four types in each

case; 2) If the soil allowable load capacity decreases, the minimum area increases; 3) The resultant

mechanical elements are R=2400 kN, MxT and MyT are equal or less than zero for all cases; 4) The

stresses generated by loads in each vertex are equal or less than σadm and greater than zero in each

case (meets the conditions indicated by the stresses). It means that the resultant force is located in

the center of gravity of the contact area of the footing with soil for the case 4 of the type 1 and case

5 of the types 1, 2, 3, because the total moments are MxT=0, MyT=0 (the stresses generated by loads

in each vertex are the same to σadm), and for the other cases the resultant force is not located in the

center of gravity of the contact area of the footing with soil, and therefore the resultant force “R”

has an eccentricity.

Table 4 shows the following results (dimensions b1 and b2 are assumed nonnegative. and

a=5.40 m and b=6.40 m): 1) The minimum area is deferent for the four types in each case; 2) If the

soil allowable load capacity decreases, the minimum area increases; 3) The resultant mechanical

elements are R=2400 kN, MxT and MyT are equal or less than zero for all cases; 4) The stresses

generated by loads in each vertex are equal or less than σadm and greater than zero in each case

(meets the conditions indicated by the stresses). It means that the resultant force is not located in

the center of gravity of the contact area of the footing with soil, and therefore the resultant force

“R” has an eccentricity for all cases.

Table 2 shows the optimal area for the dimensions a≥0, b≥0, b1≥0 and b2≥0. Table 3 presents

the optimal area for the dimensions a≥0, b≥0, b1≥1 and b2≥1. Table 4 shows the optimal area for

the dimensions a=5.40, b=6.40, b1≥0 and b2≥0. If the three tables are compared for the same cases

and types in function of the optimal area: Table 2 is smaller than Table 3, but Table 3 is smaller

than Table 4.

5. Conclusions

The foundation is an essential part of a structure that transmits column or wall loads to the

underlying soil below the structure.

The mathematical approach suggested in this paper produces results that have a tangible

accuracy for all problems, main part of this research to find the more economical dimensions of

177


corner combined footings using the optimization techniques.

The main conclusions are:

1. The most economical dimension is presented if there are fewer restricted dimensions with

respect to positive values.

2. The methodology shown in this paper is more accurate and converges more quickly.

3. The classical model will not be practical compared to this methodology, because the classical

model is developed proposing the dimensions and then verified to comply with the stresses limits

mentioned above.

The model presented in this paper is recommended for the localized columns very close

together or if the loads applied are too large in such a way that the isolated footings are overlap

between them.

The proposed model presented in this paper for dimensioning of corner combined footings

subjected to an axial load and moment in two directions in each column, also it can be applied to

others cases: 1) Footings subjected to a concentric axial load in each column, 2) Footings

subjected to a axial load and one moment in each column.

Then the proposed model is recommended for dimensioning of corner combined footings with

two continuous property line (see Table 2) and also for three and four property lines (see Table 4)

subjected to an axial load, moment around of the axis “X” and moment around of the axis “Y”

applied in each column.

The model presented in this paper applies only for dimensioning of corner combined footings

assumed than the structural member is rigid and the supporting soil layers elastic, which meet

expression of the biaxial bending, i.e., the variation of pressure is linear.

The suggestions for future research are:

1. Design for the corner combined footings assuming these are rigid and the supporting soil

layers elastic.

2. Dimensioning and design for the corner combined footings supported on another type of soil

by example in totally cohesive soils (clay soils) and totally granular soils (sandy soils), the

pressure diagram is not linear and should be treated differently.

Acknowledgments

The research described in this paper was financially supported by the Institute of

Multidisciplinary Researches of the Faculty of Accounting and Administration of the Autonomous

University of Coahuila. The authors also gratefully acknowledge the helpful comments and

suggestions of the reviewers, which have improved the presentation.

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Appendix Table 2 Results obtained by software for b1≥0 and b2≥0

Type

Resultant mechanical

elements

Optimal

area Dimension of the footing Stresses generated by loads in each vertex

R kN MxT kN-m MyT kN-m At m2 a m b m b1 m b2 m σ1 kN/m2

σ2 kN/m2 σ3 kN/m2

σ4 kN/m2 σ5 kN/m2

σ6 kN/m2

Case 1 (σadm=250.00 kN/m2)

1 2400 0.00 0.00 9.60 6.12 9.13 0.99 0.43 250.00 250.00 250.00 250.00 250.00 250.00

2 2400 0.00 0.00 9.60 8.11 8.99 0.54 0.62 250.00 250.00 250.00 250.00 250.00 250.00

3 2400 0.00 0.00 9.60 10.44 7.15 0.41 0.79 250.00 250.00 250.00 250.00 250.00 250.00

4 2400 0.00 0.00 9.60 9.19 10.46 0.56 0.45 250.00 250.00 250.00 250.00 250.00 250.00


1 2400 0.00 0.00 10.67 7.34 7.15 0.71 0.85 225.00 225.00 225.00 225.00 225.00 225.00

2 2400 0.00 0.00 10.67 7.98 8.90 0.61 0.70 225.00 225.00 225.00 225.00 225.00 225.00

3 2400 0.00 0.00 10.67 9.67 7.44 0.54 0.79 225.00 225.00 225.00 225.00 225.00 225.00

4 2400 0.00 0.00 10.67 10.25 9.26 0.48 0.65 225.00 225.00 225.00 225.00 225.00 225.00


1 2400 0.00 0.00 12.00 6.46 7.69 1.06 0.78 200.00 200.00 200.00 200.00 200.00 200.00

2 2400 0.00 0.00 12.00 7.82 8.76 0.70 0.81 200.00 200.00 200.00 200.00 200.00 200.00

3 2400 0.00 0.00 12.00 8.21 8.53 0.86 0.64 200.00 200.00 200.00 200.00 200.00 200.00

4 2400 0.00 0.00 12.00 8.73 10.56 0.77 0.54 200.00 200.00 200.00 200.00 200.00 200.00


1 2400 0.00 0.00 13.71 6.28 7.32 1.26 0.96 175.00 175.00 175.00 175.00 175.00 175.00

2 2400 0.00 0.00 13.71 6.88 7.37 1.06 0.77 175.00 175.00 175.00 175.00 175.00 175.00

3 2400 0.00 0.00 13.71 7.91 8.47 1.06 0.72 175.00 175.00 175.00 175.00 175.00 175.00

4 2400 0.00 0.00 13.71 10.47 8.64 0.57 0.96 175.00 175.00 175.00 175.00 175.00 175.00


1 2400 0.00 0.00 16.00 5.72 7.04 1.77 1.12 150.00 150.00 150.00 150.00 150.00 150.00

2 2400 0.00 0.00 16.00 7.37 8.20 1.00 1.20 150.00 150.00 150.00 150.00 150.00 150.00

3 2400 0.00 0.00 16.00 7.40 8.51 1.41 0.78 150.00 150.00 150.00 150.00 150.00 150.00

4 2400 0.00 0.00 16.00 8.70 9.73 1.00 0.84 150.00 150.00 150.00 150.00 150.00 150.00

181


Table 3 Results obtained by software for b1≥1 and b2≥1

Type

Resultant mechanical elements

Optimal area

Dimension of the footing Stresses generated by loads in each vertex


σ2 kN/m2 σ3 kN/m2

σ4 kN/m2 σ5 kN/m2

σ6 kN/m2


1 2400 -404.93 -436.54 11.44 6.04 6.40 1.00 1.00 169.13 240.54 190.41 250.00 229.65 241.47

2 2400 -862.14 -665.20 12.34 6.04 7.30 1.00 1.00 131.87 236.18 162.95 250.00 232.74 250.00

3 2400 -693.66 -837.18 12.43 7.03 6.40 1.00 1.00 131.64 234.58 161.70 250.00 230.33 250.00

4 2400 -1176.81 -1114.65 13.35 7.03 7.32 1.00 1.00 101.13 232.20 137.57 250.00 231.37 250.00


1 2400 -300.82 -281.14 11.87 6.28 6.59 1.00 1.00 176.30 218.64 189.40 225.00 218.26 225.00

2 2400 -667.06 -515.91 12.91 6.31 7.60 1.00 1.00 142.00 215.61 163.05 225.00 213.33 225.00

3 2400 -565.96 -647.79 12.95 7.32 6.63 1.00 1.00 140.74 213.81 161.92 225.00 215.01 225.00

4 2400 -982.82 -931.38 13.98 7.34 7.63 1.00 1.00 112.30 212.01 138.87 225.00 211.42 225.00


1 2400 -111.86 -104.62 12.48 6.58 6.90 1.00 1.00 183.73 197.96 187.94 200.00 197.84 200.00

2 2400 -439.64 -340.97 13.57 6.62 7.95 1.00 1.00 150.86 194.65 162.82 200.00 193.38 200.00

3 2400 -377.74 -431.73 13.62 7.66 6.96 1.00 1.00 149.59 193.58 161.75 200.00 194.26 200.00

4 2400 -756.01 -716.86 14.71 7.70 8.00 1.00 1.00 122.28 191.41 139.85 200.00 191.03 200.00


1 2400 0.00 0.00 13.71 6.37 7.20 1.21 1.00 175.00 175.00 175.00 175.00 175.00 175.00

2 2400 -169.55 -131.93 14.37 6.99 8.38 1.00 1.00 158.20 173.25 162.10 175.00 172.85 175.00

3 2400 -153.17 -174.75 14.42 8.08 7.34 1.00 1.00 156.93 172.81 161.09 175.00 173.03 175.00

4 2400 -485.70 -460.87 15.58 8.13 8.45 1.00 1.00 130.86 170.35 140.36 175.00 170.14 175.00


1 2400 0.00 0.00 16.00 5.72 7.04 1.77 1.12 150.00 150.00 150.00 150.00 150.00 150.00

2 2400 0.00 0.00 16.00 7.06 8.50 1.11 1.10 150.00 150.00 150.00 150.00 150.00 150.00

3 2400 0.00 0.00 16.00 8.31 7.39 1.07 1.13 150.00 150.00 150.00 150.00 150.00 150.00

4 2400 -155.49 -147.67 16.65 8.66 8.99 1.00 1.00 137.73 148.78 140.23 150.00 148.72 150.00

182


Table 4 Results obtained by software for a=5.40 and b=6.40

Type

Resultant mechanical elements

Optimal area

Dimension of the footing

Stresses generated by loads in each vertex


σ2 kN/m2 σ3 kN/m2

σ4 kN/m2 σ5 kN/m2

σ6 kN/m2


1 2400 -562.10 -502.51 11.89 5.40 6.40 1.32 0.94 146.52 231.78 179.59 250.00 235.15 250.00

2 2400 -1073.17 -830.21 13.91 5.40 6.40 1.25 1.39 81.98 224.41 144.27 250.00 213.30 250.00

3 2400 -927.87 -985.36 14.50 5.40 6.40 2.04 0.80 68.49 198.43 139.42 250.00 230.64 250.00

4 2400 -1491.83 -1344.31 16.46 5.40 6.40 2.00 1.28 21.35 191.08 120.64 250.00 209.63 250.00


1 2400 -471.98 -421.88 12.85 5.40 6.40 1.44 1.02 142.35 209.24 170.77 225.00 212.34 225.00

2 2400 -952.61 -738.92 15.05 5.40 6.40 1.35 1.54 82.17 202.04 139.21 225.00 190.92 225.00

3 2400 -817.15 -861.13 15.68 5.40 6.40 2.26 0.84 70.30 176.14 135.59 225.00 208.57 225.00

4 2400 -1358.36 -1213.51 17.84 5.40 6.40 2.23 1.39 25.65 168.33 119.05 225.00 188.27 225.00


1 2400 -366.72 -327.49 13.96 5.40 6.40 1.59 1.12 138.98 187.32 161.66 200.00 190.00 200.00

2 2400 -810.79 -632.51 16.36 5.40 6.40 1.47 1.71 83.32 180.27 133.73 200.00 169.32 200.00

3 2400 -689.05 -713.66 17.01 5.40 6.40 2.53 0.86 73.43 155.09 131.40 200.00 187.02 200.00

4 2400 -1203.77 -1057.16 19.41 5.40 6.40 2.51 1.50 31.13 146.27 116.89 200.00 167.97 200.00


1 2400 -242.80 -216.32 15.24 5.40 6.40 1.77 1.23 136.65 166.25 152.12 175.00 168.27 175.00

2 2400 -643.34 -507.74 17.85 5.40 6.40 1.60 1.92 85.71 159.25 127.63 175.00 148.83 175.00

3 2400 -540.85 -540.38 18.48 5.40 6.40 2.86 0.85 78.39 135.83 126.63 175.00 165.93 175.00

4 2400 -1024.52 -868.40 21.19 5.40 6.40 2.87 1.61 38.14 125.21 113.89 175.00 149.04 175.00


1 2400 -95.82 -84.93 16.73 5.40 6.40 1.99 1.36 135.64 146.37 141.96 150.00 147.30 150.00

2 2400 -446.25 -360.53 19.56 5.40 6.40 1.73 2.18 89.70 139.07 120.60 150.00 130.02 150.00

3 2400 -370.32 -344.97 20.04 5.40 6.40 3.24 0.80 85.73 120.00 120.81 150.00 144.93 150.00

4 2400 -816.14 -639.89 23.19 5.40 6.40 3.34 1.68 47.20 105.87 109.61 150.00 131.72 150.00

183

Optimal dimensioning for the corner combined footings

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